The Qth-power algorithm in characteristic 0

The Qth-power algorithm produces a useful canonical P-module presentation for the integral closures of certain integral extensions of $P:=\mathbf{F}[x_n,\mathrm{.}\mathrm{.}\mathrm{.},x_1]$, a polyonomial ring over the finite field $\mathbf{F}:={\mathbf{Z}}_q$ of $q$ elements. Here it is shown how to use this for several small primes $q$ to reconstruct similar integral closures over the rationals $\mathbf{Q}$ using the Chinese remainder theorem to piece together presentations in different positive characteristics, and the extended Euclidean algorithm to reconstruct rational fractions to lift these to presentations over $\mathbf{Q}$.